variable speed generatorsch (2)

1 Wound Rotor Induction Generators (WRIGs): Steady State 1.1 1.2 Introduction 1-1 Construction Elements 1-4 Magnetic Cores • Windings and Their mmfs • Slip-Rings and Brushes 1.3 1.4 1.5 1.6 Steady-State Equations 1-9 Equivalent Circuit 1-11 Phasor Diagrams 1-13 Operation at the Power Grid 1-18 Stator Power vs Power Angle • Rotor Power vs Power Angle • Operation at Zero Slip (S = 0) 1.7 Autonomous Operation of WRIG 1-22 1.8 Operation of WRIG in the Brushless Exciter Mode 1-28 1.9 Losses and Efficiency of WRIG 1-33 1.10 Summary 1-34 References 1-36 1.1 Introduction Wound rotor induction generators (WRIGs) are provided with three phase windings on the rotor and on the stator They may be supplied with energy at both rotor and stator terminals This is why they are called doubly fed induction generators (DFIGs) or double output induction generators (DOIGs) Both motoring and generating operation modes are feasible, provided the power electronics converter that supplies the rotor circuits via slip-rings and brushes is capable of handling power in both directions As a generator, the WRIG provides constant (or controlled) voltage Vs and frequency f1 power through the stator, while the rotor is supplied through a static power converter at variable voltage Vr and frequency f2 The rotor circuit may absorb or deliver electric power As the number of poles of both stator and rotor windings is the same, at steady state, according to the frequency theorem, the speed ωm is as follows: ω m = ω1 ± ω ; ω m = Ω R ⋅ p1 (1.1) where p1 is the number of pole pairs ΩR is the mechanical rotor speed 1-1 © 2006 by Taylor & Francis Group, LLC 1-2 Variable Speed Generators Slip rings Prime mover WRIG ωm Brushes f2, Vr - variable Pm input Bidirectional a.c –a.c static converter WRIG wm = w1 – w2 < w1 w2 > Mechanical Power power f1 = ct Vs = ct Pr Trafo Rotor electric power input Pm = ∑losses + Ps – Pr ~ f1, Vs - constant (a) (b) Input Pm Input Mechanical power Ps Stator electric Ps Stator electric WRIG wm = w1 – w2 > w1 w2 < Power f1 = ct Vs = ct Mechanical power Pm WRIG wm = w1 + wm w2 > w1 Ps Stator electric Power f1 – ct Vs – variable Rotor electric power output Pr Pr Vr - variable f2 > f1 - variable Rotor electric power input Pm = ∑losses – Ps + Pr Pm = ∑losses + Ps + Pr (d) (c) FIGURE 1.1 Wound rotor induction generator (WRIG) main operation modes: (a) basic configuration, (b) subsynchronous generating (ωr < ω1), (c) supersynchronous generating (ωr > ω1), and (d) rotor output WRIG (brushless exciter) The sign is positive (+) in Equation 1.1 when the phase sequence in the rotor is the same as in the stator and ωm < ω1, that is, subsynchronous operation The negative (−) sign in Equation 1.1 corresponds to an inverse phase sequence in the rotor when ωm > ω1, that is, supersynchronous operation For constant frequency output, the rotor frequency ω2 has to be modified in step with the speed variation This way, variable speed at constant frequency (and voltage) may be maintained by controlling the voltage, frequency, and phase sequence in the rotor circuit It may be argued that the WRIG works as a synchronous generator (SG) with three-phase alternating current (AC) excitation at slip (rotor) frequency ω2 = ω1 − ωm However, as ω1 ≠ ωm, the stator induces voltages in the rotor circuits even at steady state, which is not the case in conventional SGs Additional power components thus occur The main operational modes of WRIG are depicted in Figure 1.1a through Figure 1.1d (basic configuration shown in Figure 1.1a) The first two modes (Figure 1.1b and Figure 1.1c) refer to the already defined subsynchronous and supersynchronous generations For motoring, the reverse is true for the rotor circuit; also, the stator absorbs active power for motoring The slip S is defined as follows: S= © 2006 by Taylor & Francis Group, LLC ω > 0; subsynchronous operation ω1 < 0; supersynchronous operation e (1.2) Wound Rotor Induction Generators (WRIGs): Steady State 1-3 A WRIG works, in general, for ω2 ≠ (S ≠ 0), the machine retains the characteristics of an induction machine The main output active power is delivered through the stator, but in supersynchronous operation, a good part, about slip stator powers (SPs), is delivered through the rotor circuit With limited speed variation range, say from Smax to −Smax, the rotor-side static converter rating — for zero reactive power capability on the rotor side — would be Ρ conv ≈ | Smax | Ps With Smax typically equal to ±0.2 to 0.25, the static power converter ratings and costs would correspond to 20 to 25% of the stator delivered output power At maximum speed, the WRIG will deliver increased electric power, Pmax: Pmax = Ps + Prmax = Ps + Smax Ps (1.3) with the WRIG designed at Ps for ωm = ω1 speed The increased power is delivered at higher than rated speed: ω mmax = ω1(1+ | Smax |) (1.4) Consequently, the WRIG is designed electrically for Ps at ωm = ω1, but mechanically at wmmax and Pmax The capability of a WRIG to deliver power at variable speed but at constant voltage and frequency represents an asset in providing more flexibility in power conversion and also better stability in frequency and voltage control in the power systems to which such generators are connected The reactive power delivery by WRIG depends heavily on the capacity of the rotor-side converter to provide it When the converter works at unity power delivered on the source side, the reactive power in the machine has to come from the rotor-side converter However, such a capability is paid for by the increased ratings of the rotor-side converter As this means increased converter costs, in general, the WRIG is adequate for working at unity power factor at full load on the stator side Large reactive power releases to the power system are still to be provided by existing SGs or from WRIGs working at synchronism (S = 0, ω2 = 0) with the back-to-back pulse-width modulated (PWM) voltage converters connected to the rotor controlled adequately for the scope Wind and small hydroenergy conversion in units of megawatt (MW) and more per unit require variable speed to tap the maximum of energy reserves and to improve efficiency and stability limits High-power units in pump-storage hydro- (400 MW [1]) and even thermopower plants with WRIGs provide for extra flexibility for the ever-more stressed distributed power systems of the near future Even existing (old) SGs may be retrofitted into WRIGs by changing the rotor and its static power converter control The WRIGs may also be used to generate power solely on the rotor side for rectifier loads (Figure 1.1d) To control the direct voltage (or direct current [DC]) in the load, the stator voltage is controlled, at constant frequency ω1, by a low-cost alternating current (AC) three-phase voltage changer As the speed increases, the stator voltage has to be reduced to keep constant the current in the DC load connected to the rotor (ω2 = ω1 + ωm) If the machine has a large number of poles (2p1 = 6,8,12), the stator AC excitation input power becomes rather low, as most of the output electric power comes from the shaft (through motion) Such a configuration is adequate for brushless exciters needed for synchronous motors (SMs) or for generators, where field current is needed from zero speed, that is, when full-power converters are used in the stator of the respective SMs or SGs With 2p1 = 8, n = 1500 rpm, and f1 = 50 Hz, the frequency of the rotor output f2 = f1 + np1 = 50 + (1500/60)∗ = 150 Hz Such a frequency is practical with standard iron core laminations and reduces the contents in harmonics of the output rectified load current In this chapter, the following subjects related to WRIG steady state will be detailed: • • • • Construction elements Basic principles Inductances Steady-state model (equations, phasor diagram, equivalent circuits) © 2006 by Taylor & Francis Group, LLC 1-4 Variable Speed Generators • Steady-state characteristics at power grid • Steady-state characteristics for isolated loads • Losses and efficiency 1.2 Construction Elements The WRIG topology contains the following main parts: • • • • • • • Stator laminated core with Ns uniformly distributed slots Rotor laminated core with Nr uniformly distributed slots Stator three-phase winding placed in insulated slots Rotor shaft Stator frame with bearings Rotor copper slip-rings and stator (placed) brushes to transfer power to (from) rotor windings Cooling system 1.2.1 Magnetic Cores The stator and rotor cores are made of thin (typically 0.5 mm) nonoriented grain silicon steel lamination provided with uniform slots through stamping (Figure 1.2.a) To keep the airgap reasonably small, without incurring large core surface harmonics eddy current losses, only the slots on one side may be open On the other side of the airgap, they should be half closed or half open (Figure 1.2b) Though, in general, the use of radial–axial ventilation systems led to the presence of radial channels between 60 and 100 mm long elementary stacks, at least for powers up to to MW, axial ventilation with single lamination stacks is feasible (Figure 1.3a and Figure 1.3b) As the airgap is slightly increased in comparison with standard induction motors, the axial airflow through the airgap is further facilitated The axial channels (Figure 1.3a) in the stator and rotor yokes (behind the slot region) play a key role in cooling the stator and the rotor, as the radial channels (Figure 1.3b) for the radial–axial ventilation The radial channels, however, are less efficient, as they are “traveled” by the windings, and thus, additional phase resistance and leakage inductance are added by the winding zones in the radial channel contributions In very large, or long, stack machines, radial–axial cooling may be inevitable, but, as explained before, below MW, the axial cooling in unistack cores, already in industrial use for induction motors, seems to be the way of the future Open stator slot g Semiopen rotor slot (a) FIGURE 1.2 (a) Stator and (b) rotor slotted lamination © 2006 by Taylor & Francis Group, LLC (b) 1-5 Wound Rotor Induction Generators (WRIGs): Steady State Air flow Axial channels (air flow) Airgap Air flow Radial channels (air flow) Airgap Air flow Axial channels (air flow) Air flow (a) Air flow (b) FIGURE 1.3 Stator and rotor stacks: (a) for axial cooling and (b) for radial–axial cooling 1.2.2 Windings and Their mmfs The stator and rotor three-phase windings are similar in principle In Chapter in Synchronous Generators, their design is described in some detail Here, only the basic issues are presented The three-phase windings are built to provide for traveling magnetomotive forces (mmfs) capable of producing a traveling magnetic field in the uniform airgap (slot openings are neglected or considered through the Carter coefficient KC = 1.02 to 1.5): Bg ( x , t ) = µo Fs ,r (x , t ) gK C (1 + K s ) (1.5) where Fs,r(x,t) is equal to the mmfs per pole produced by either stator or rotor windings g is the airgap KC is the Carter coefficient to account for airgap increase due to slot openings Ks is the iron core contribution to equivalent magnetic reluctance of the main flux path (Figure 1.2a) To produce a traveling airgap field, the stator and rotor mmfs, seen from the stator and from the rotor, respectively, have to be as follows: Fs (θ s , t ) = F1s cos( p1θ s − ω1t ) (1.6) Fr (θr , t ) = F1r cos( p1θr ± ω 2t ) (1.7) where p1 is the number of electrical periods of the magnetic field wave in the airgap or of pole pairs The rotor mmf is produced by currents of frequency ω2 At constant speed, the rotor and stator geometrical angles are related by p1θr = p1θ s − ω r t + γ ; ω r = Ωr ⋅ p1 ; p1θ s = ω1t (1.8) where ωr is the rotor speed in electrical radians per second (rad/sec) Consequently, Fr(θs ,t) becomes Fr (θ s , t ) = F1 cos[ p1θ s − (ω r ± ω )t + γ ] © 2006 by Taylor & Francis Group, LLC (1.9) 1-6 Variable Speed Generators The average electromagnetic torque and power per electric period is nonzero only if the two mmfs are at standstill with each other That is, ω = ωr ± ω ; S = ω / ω (1.10) The positive sign (+) is used when ωr < ω1, and thus, the rotor and stator mmf waves rotate in a positive direction The negative sign (−), used when ωr > ω1, refers to the case when the rotor mmf wave moves in the opposite direction to that of the stator Also, the torque is nonzero when the angle γ ≠ 0, that is, when the two mmfs are phase shifted To produce a traveling mmf, three phases, space lagged by 120° (electrical), have to be supplied by AC currents with 120° (electrical) time-lag angles between them (see Chapter in Synchronous Generators, on the SG) So, all three phase windings for, say, maximum value of current, should independently produce a sinusoidal spatial mmf: (FsA,B ,C (θ s , t )) t =0  2π  = F1s cos  p1θ s − (i − 1)  3   (1.11) Each phase mmf has to produce 2p1 semiperiods along a mechanical period With only one coil per pole per phase, there would be 2p1 coils per phase and 2p1 slots per phase if each coil occupies half of the slot (Figure 1.4a) From the rectangular distribution of phase mmf (Figure 1.3a and Figure 13.b), a fundamental is extracted: FsA ( p1 ,θ s ) = n I cos p1θ s ; π c ns − turns/coil (1.12) The harmonics content of the phase mmf in Figure 1.4b is hardly acceptable, but more steps in its distribution (more slots) and chorded coil would drastically reduce these space harmonics (Figure 1.5) For the two-pole 24-slot winding with chorded coils (coil span/pole pitch = 10/12), the number of steps in the phase mmf is larger, and thus, the harmonics are reduced (Figure 1.5) For the fundamental component (based on Figure 1.5b), we obtain the expression of the mmf per pole and phase: S N X A N X S 10 11 (a) 12 X nc I√2 p1qs (b) nc I√2 FsA(p1qs) 2π 4π FIGURE 1.4 Elementary three-phase winding with 2p1 = poles and Ns = 12 slots: (a) coils of phase A in series and (b) phase A magnetomotive force (mmf) for maximum phase current © 2006 by Taylor & Francis Group, LLC 1-7 Wound Rotor Induction Generators (WRIGs): Steady State y (coil span) From 15 To to From 16 10 11 12 13 14 15 16 17 18 19 20 21 22 23 A 24 X (a) nc I√2 ncI√2 t (pole pitch) (b) FIGURE 1.5 Two-pole (2p1 = 2), Ns = 24 slots three-phase winding, with two layers in slot, coil span y/τ = 10/12: (a) slotto-phase allocation for layer and coils of phase A and (b) phase A magnetomotive force (mmf) for maximum current FbA1 = 2W1kW KY ⋅ I π p1 ; W1 − turns/phase (1.13) For the space harmonic ν, in a similar way, FsAν = 2W1kdν KY ν ⋅ I ν p1 (1.14) with Kdn and Kyn known as distribution and chording factors: K dν = sin νπ /6 νπ q sin 6q K yν = sin νy π τ (1.15) where q is the number of slots per pole per phase: qs ,r = N s,r p1m1 = N s,r p1 (1.16) Only the odd harmonics are present, in general, as the positive and negative mmf poles are identical, while the multiples of three harmonics are zero for symmetric currents (equal amplitude, 120° phase shift): ν = 1,5,7,11,13,17,19,… It was proven (Chapter 4, in Synchronous Generators) that harmonics 7,13,19 are positive, and 5,11,17,… are negative in terms of sequence By adding the contributions of the three phases, we find that the mmf amplitude per pole Fsn is as follows: 3W1K dν KY ν I Fsν = FsAν = ν p1 © 2006 by Taylor & Francis Group, LLC (1.17) 1-8 Variable Speed Generators Similar expressions may be derived for the rotor To avoid parasitic synchronous torques, the number of slots of the stator and the rotor has to differ: N s ≠ Nr ; qs ≠ qr (1.18) Harmonics have to be treated carefully, as the radial magnetic pull due to rotor excentricity tends to be larger in WRIG than in cage-rotor induction generators (IGs) [2] In general, WRIGs tend to be built with integer q both in the stator and in the rotor Also, current paths in parallel may be used to reduce elementary conductor cross-sections Frequency (skin) effects have to be reduced, especially in large WRIGs, with bar-made windings where transposition may be necessary (Roebel bar, see Chapter 7, in Synchronous Generators) Finally, the rotor winding end connections have to be protected against centrifugal forces through adequate bandages, as for cylindrical rotor SGs Whenever possible, the rated (design) voltage of the rotor winding has to be equal to that in the stator as required in the control of the rotor-side static power converter at maximum slip This way, a voltagematching transformer is avoided on the supply side of the static converter Consequently, the rotor-tostator turns ratio ars is as follows: ars ≅ r r Wr K q1 K d1 s q1 Ws K K s d1 ≅ |Smax| (1.19) Care must be exercised in such designs to avoid connecting the stator at the full-voltage power grid at zero speed (S = 1), as the voltage induced in the rotor windings will be ars times larger than the rated one, jeopardizing the rotor winding insulation and the rotor-side static power converter If starting as a motor is required (for pump storage, etc.), it is done from the rotor, with the stator short-circuited, by making use of the rotor-side bidirectional power flow capabilities Then, at certain speed ωrmin > ωrn(1 − |Smax|), the stator circuit is opened The machine is cruising while the control prepares the synchronization conditions by using the inverter on the rotor to produce adequate voltages in the stator After synchronization, motoring (for pump storage) can be performed safely In WRIGs, a considerable amount of power (up to |Smax|•PsN) is transferred in and out of the rotor electrically through slip-rings and brushes With |Smax| = 0.20, it is about 20% of the rated power of the machine Remember that in SGs, the excitation power transfer to rotor by slip-rings and brushes is about five to ten times less The question is if those multimegawatts may be transferred through slip-rings and brushes to the rotor in large-power WRIGs The answer seems to be “yes,” as 200 MW and 400 MW units have been in operation for more than years at up to 30 MW power transfer to the rotor In contrast to SGs, WRIGs have to use higher voltage for the power transfer to the rotor to reduce the slip-ring current Multilevel voltage source bidirectional pulse-width modulated (PWM) MOSFETcontrolled thyristor (MCT) converters are adequate for the scope of our discussion here If the rotor voltage is increased in the kilovolt (and above) range, the insulation provisions for the rotor slip-rings and on the brush framing side are much more demanding Note that SG brushless exciters based on the WRIG principle with rotor rectified output not need slip-rings and brushes In WRIGs with large stator voltage (Vn = 18 kV, 400 MW), it may be more practical to use lower rated (maximum) voltage in the rotor, say up to 4.5 kV, and then use a step-up voltage adapting transformer to match the rotor connected static power converter voltage (4.5 kV) to the local (stator) voltage (say 18 kV) Such a reduction in voltage may reduce the eventual costs of the static power converter so much as to overcompensate the costs of the added transformer 1.2.3 Slip-Rings and Brushes A typical slip-ring rotor is shown in Figure 1.6 It is obvious that three copper rings serve each phase, as the rotor currents are large © 2006 by Taylor & Francis Group, LLC Wound Rotor Induction Generators (WRIGs): Steady State 1-9 FIGURE 1.6 Slip-ring wound rotor 1.3 Steady-State Equations The electromagnetic force (emf) self-induced by the stator winding, with the rotor winding open, E1, is as follows: E1 = π f1W1KW 1φ10 ; (RMS) (1.20) KW = K d1 ⋅ K y1 (1.21) B τl π g10 i (1.22) The flux per pole φ10 is φ10 = where li is the stack length τ is the pole pitch Dis is the stator bore diameter Bg10 is the airgap fundamental flux density peak value: Bg10 = µo Fs10 K C g (1 + K s ) (1.23) F1so is the amplitude of stator mmf fundamental per pole From Equation 1.17, with ν = 1, Fs10 = © 2006 by Taylor & Francis Group, LLC 3W1KW 1I o π p1 (1.24) 1-10 Variable Speed Generators Bg10 (T) τ/g increases 1.0 0.8 τ/g increases L1m = l1m LN LN = 0.6 VN IN w N 0.4 0.2 I10 IN I10 IN 0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 FIGURE 1.7 Typical airgap flux density (Bg10) and magnetization inductance (in per unit [P.U.]) vs P.U stator current But the same emf E1 may be expressed as E1 = ω1L1m ⋅ I10 (1.25) So, the main flux, magnetization (cyclic) inductance of the stator — with all three phases active and symmetric — L1m is as follows (from Equation 1.20 through Equation 1.25): L1m = µ0 (W1KW 1)2 τ li π p1K C g (1 + K s ) (1.26) The Carter coefficient KC > accounts for both stator and rotor slot openings (KC ≈ KC1KC2) The saturation factor KS, which accounts for the iron core magnetic reluctance, varies with stator mmf (or current for a given machine), and so does magnetic inductance L1m (Figure 1.7) Besides L1m, the stator is characterized by the phase resistance Rs and leakage inductance Lsl [2] The same stator current induces an emf E2s in the rotor open-circuit windings With the rotor at speed ωr — slip S = (ω1 − ωr)/ω1 — E2s has the frequency f2 = Sf1: E2 s (t ) = E2 s cosω 2t E2 s = π 2Sf1W2 KW 2φ10 (1.27) Consequently, E2 s WK = S W = S ⋅ K rs E1 W1KW (1.28) This rotor emf at frequency Sf1 in the rotor circuit is characterized by phase resistance Rrr and leakage inductance Lrrl Also, the rotor is supplied by a system of phase voltages at the same frequency ω2 and at a prescribed phase The stator and rotor equations for steady-state/phase may be written in complex numbers at frequency ω1 in the stator and ω2 in the rotor: (Rs + jω1Lsl )I s − V s = E1 at ω1 (1.29) (Rrr + jSω1 Lrl )I r − V r = E s at ω (1.30) r © 2006 by Taylor & Francis Group, LLC r 1-22 Variable Speed Generators Similar graphs Pr r (δ − δ k )and Qr r (δ − δ k )may be drawn by using these expressions, but they are of a smaller practical use than Ps and Qs They are, however, important for designing the rotor-side static power converter and for determining the total rotor electric power delivery, or absorption, during subsynchronous or supersynchronous operation 1.6.3 Operation at Zero Slip (S = 0) At zero slip, from Equation 1.62, it follows first that δ k = π/2 Finally, from Equation 1.59 and Equation 1.60, Ps = −3VsVr Qs = 3Vs ω1 Ls −3  π sin  δ +  Rr Ls 2  Lm VsVr Lm Rr Ls Ir =  π cos  δ +  2  Vr Rr (1.70) (1.71) (1.72) Note again that the rotor voltage is considered in stator coordinates The power angle (δ + π ) is typical for SGs, where it is denoted by δ u (the phase shift between rotor-induced emf and the phase voltage) For operation at zero slip (S = 0), when the rotor circuit is DC fed, all the characteristics of SGs hold true In fact, it seems adequate to run the WRIG at S = when massive reactive power delivery (or absorption) is required Though active and reactive power capability circles may be defined for WRIG, it seems to us that, due to decoupled fast active and reactive power control through the rotor-connected bidirectional power converters (Chapter 8, in Synchronous Generators), such graphs may become somewhat superfluous 1.7 Autonomous Operation of WRIG Insularization of WRIGs, in case of need, from the power grids, caused by excess power in the system or stability problems, leads to autonomous operation Autonomous operation is characterized by the fact that voltage has to be controlled, together with stator frequency (at various rotor speeds in the interval [1 ± |Smax|]), in order to remain constant under various active and reactive power loads Whatever reactive power is needed by the consumers, it has to be provided from the rotor-side converter after covering the reactive power required to magnetize the machine When large reactive power loads are handled, it seems that running at constant speed and zero slip (S = 0) would be adequate for taking full advantage of the rotor-side static converter limited ratings and for limiting rotor windings and converter losses On the other hand, for large active loads, supersynchronous operation is suitable, as the WRIG may be controlled to operate around unity power factor while keeping the stator voltage within limits Subsynchronous operation should be used when part loads are handled in order to provide for better efficiency of the prime mover for partial loads The equivalent circuit (Figure 1.8) may easily be adapted to handle autonomous loads under steady state (Figure 1.14) For autonomous operation, the stator voltage Vs is replaced by the following: V s = −(RLoad + jX Load )I s (1.73) In these conditions, retaining the power angle δ as a variable does not seem to be so important The rotor voltage “sets the tone” and may be considered in the real axis: V r = Vr Neglecting the stator © 2006 by Taylor & Francis Group, LLC 1-23 Wound Rotor Induction Generators (WRIGs): Steady State Is jω1Ls1 Rs jω1Lr1 Rr/S Ir Im RLoads R1m Vr Vs S jXLoads jω1Lm FIGURE 1.14 Equivalent circuit of wound rotor induction generator (WRIG) for autonomous operation resistance Rs does not bring any simplification, as it is seen in series with the load (Equation 1.73): [Rs + RLoad + j( X Load + X sl )]I s = − jX1m (I s + I r ) = E m (I m ) (Rs + jSX rl )I r − V r = − jSX1m (I s + I r ) = E m ⋅ S; Im = I s + Ir (1.74) Both equations are written in stator coordinates (at frequency ω1 for all reactances) We may consider now the WRIG as being supplied only from the rotor, with the stator connected to an external impedance In other words, the WRIG becomes a typical induction generator fed through the rotor, having stator load impedance It is expected that such a machine would be a motor for positive slip (S > 0, ωr < ω1) and a generator for negative slip (S < 0, ωr > ω1) This is a drastic change of behavior with respect to the WRIG connected at a fixed frequency and voltage (strong) power grid, where motoring and generating are practical both subsynchronously and supersynchronously By properly adjusting the rotor frequency ω2 with speed ωr to keep ω1 constant and controlling the amplitude and phase sequence of rotor voltage Vr , the stator voltage may be kept constant until a certain stator current limit, for given load power factor, is reached To obtain the active and reactive powers of the stator and the rotor Ps, Qs, Prr, Qrr, solving first for the stator and rotor currents in Equation 1.74 is necessary Neglecting the core loss resistance R1m (R1m = 0) yields the following: Is = Vr Rse (S) + jX se (S) Rse (S) = X se (S) = Rr X s +l − SX r X s +l X1m SX r Rs +l + X s +l Rr X1m Rs +l = Rs + Rloads ; X s = X sl + X1m ; + SX1m (1.75) X s +l = X s + X loads X r = X rl + X1m The active and reactive powers of stator and rotor are straightforward: Ps = 3I s2 RLoads > S < generating Qs = 3I s2 X Loads < > © 2006 by Taylor & Francis Group, LLC S > motoring (1.76) 1-24 Variable Speed Generators Also, I r from Equation 1.74 is Ir = j (Rs +l + jX s +l )I s X1m ( Pr r + jQr r = Vr ⋅ Ir ∗ (1.77) ) (1.78) The mechanical power Pm is simply Pm = 1− S  3R I − Pr r   S  r r ( (1.79) ) Qr r − X sl I s + X sl Ir + X1m Im = Qs (1.80) As the machine works as an induction machine fed to the rotor, with passive impedance in the stator, all characteristics of it may be used to describe its performance The power balance for motoring and generating is described in Figure 1.15a and Figure 1.15b Note that subsynchronous operation as a motor is very useful when self-starting is required The stator is short-circuited (Rload = Xload = 0), and the machine accelerates slowly (to observe the rotorside converter Plow rating) until it reaches the synchronization zone ωr(1 ± |Smax|) Then the stator circuit is opened, but the induced voltage in the stator has a small frequency Consequently, the phase sequence in the rotor voltages has to be reversed to obtain ω1 > ωr for the same direction of rotation This is the beginning of the resynchronization control mode when the machine is free-wheeling Finally, within a few milliseconds, the stator voltage and frequency conditions are met, and the machine stator is reconnected to the load Induction motoring with a short-circuited stator is useful for limited motion during bearing inspections or repairs (Motoring) Pload (electric) Pm (mechanical) r Pr (electric) ∑p losses (a) (Generating) Pload (electric) Pm (mechanical) ∑p losses Sa < r Pr (electric) (b) FIGURE 1.15 Power balance: (a) Sa > and (b) Sa < © 2006 by Taylor & Francis Group, LLC Wound Rotor Induction Generators (WRIGs): Steady State 1-25 Autonomous generating (on now-called ballast load) may be used as such and when, after load rejection, fast braking of the mover is required to avoid dangerous overspeeding until the speed governor takes over The stator voltage regulation in generating may be performed through changing the rotor voltage amplitude while the frequency ω1 is controlled to stay dynamically constant by modifying frequency ω2 in the rotor-side converter Example 1.2 For the WRIG in Example 1.1 at S = −0.25, f1 = 50 Hz, Is = IsN /2 = 602 A, Vr = Vrmax = Vs, cos ϕs = 1, compute the following: • • • • The load resistance Rloads per phase in the stator The load (stator voltage) Vs and load active power Ps The rotor current and active and reactive power in the rotor Pr, Qr The no-load stator voltage for this case and the phasor diagram After the computations are made, discuss the results Solution • We have to go straight to Equation 1.75, with, Rs = Rr = 0.018 Ω, X sl = X rl = 0.018 Ω, • X lm = 14.4 Ω, X s = X r = 14.58 Ω, Vsphase = 6000/ V , S = −0.25, I s = 602 A, X load = (cosϕ s = 1), where the only unknown is Rloads:  0.018  (−0.25 ⋅14.88⋅14.88) ⋅ Rse (S) =  + Rloads  − = 3.69 + 1.25 ⋅10−3 ⋅ Rloads  0.018  14.4   (14.58⋅ 0.018) 14.58 ⋅ X se (S) =  −0.25 ⋅ + Rloads  − = −3.5938 − 0.253 ⋅ Rloads   0.018 14.4 I s = 602 = Vr Rse (S) + jX se (S) = (3.69 + 1.25 ⋅10 6000/ −3 ) Rload − j(3.594 − +0.253R load ) Consequently, Rloads ≈ 3.276 Ω • The stator voltage per phase Vs is simply (cosϕ s = 1) (Vs )phase = Rload I s = 3.276 ⋅ 602 = 1972 V Ps = −3Rloads I s = −3 ⋅ 3.276 ⋅ 6022 = −3.5617 MW • The rotor current (Equation 1.77) is Ir = j (Rs + Rloads + jX s +l )I s X1m with I s = 602 ⋅(0.641 + j 0.767) So, Ir = j © 2006 by Taylor & Francis Group, LLC (0.018 + 3.276 + j14.58) ⋅ 602 ⋅ (0.641 + j 0.767) = 624.86∠+ 217 14.4 1-26 Variable Speed Generators The active and reactive powers in the rotor are as follows: Pr r + jQr r = 3Vr Ir∗ = ⋅ 6000 ⋅ 624(−0.79 + j 0.6018) = −5.116 MW + j 3.907 MVAR So, the rotor circuit absorbs reactive power to magnetize the machine, but it delivers active power, together with the stator The mechanical power covers for all losses in the machine and produces both Ps and Qs: Pm − Σp = | Ps + Pr | = −3.5617 − 5.116 = −8.6777 MW The losses considered in our example are only the winding losses: Σp = 3Rs I s + 3Rr Ir = ⋅ 0.018 ⋅(6022 + 624.862 ) = 40.654 KW So, the mechanical power is as follows: Pm = 8.6717 + 0.04654 = 8.712 MW • The no-load voltage in the stator for the above conditions is simply Em = X1m Im ; Im = I s + Ir I m = 602∠50.13° + 624.86∠217° = −107.75 + j 85.69 This is the magnetization current for the airgap flux: Em = X1m Im = 14.4 ⋅137.669 = 1982.4 V The voltage regulation is very small: ∆V = Em − Vs 1982.4 − 1972 = = 0.5246% Vs 1982.2 The current and voltage phasors are shown in Figure 1.16 Discussion To force the delivery of notable active power from the machine, we considered Vrmax = Vsn = 3.468 V ; the trouble is that Vr is reduced to the stator, and thus, for our case, when the turns ratio is defined by 1/Smax = 4.0, the actual rotor voltage meant by Vr = Vs would, in fact, imply Vrr = 4.0 Vr = 4.6000/ = 13.872 kV /phase In contrast, when the same machine (Example 1.1) delivered the power Ps = 12.5 MW through the stator (rated) plus 2.99 MW through the rotor at S = −0.25, f1 = 50 Hz , the rotor voltage Vr was only Vr = 847 V With the same rotor/stator turn ratio of 4.0, the actual rotor voltage would be, in this latter case, Vr ′ = 847 ⋅ = 3388 V , which is very close to the rated stator voltage, as intended from the start © 2006 by Taylor & Francis Group, LLC 1-27 Wound Rotor Induction Generators (WRIGs): Steady State Is Im 217° 50° VR Ir V s = –Rloads I s FIGURE 1.16 Phasors for autonomous wound rotor induction generator (WRIG) operation at S = −0.25, f1 = 50 Hz To reduce the rotor output power and voltage Vr and increase the stator output, the slip has to be reduced drastically Let us consider Is′ = 700 A and Vr = 847 V (as it was in Example 1.1) but for S = −0.05 Repeating the calculations as above, we obtain Rloads ≈ 5.734 Ω The stator voltage Vs′ is as follows: Vs ′ = I s′Rload = 700 ⋅ 5.734 = 4.0 kV /phase The stator power Ps = −3Rloads ⋅ I s′2 = 3Vs′I s′ = ⋅ ⋅103 ⋅ 700 = −8.40 MW The stator and rotor currents are thus, Is = Ir = I s j Vr 847 = Rse (s) + jX se (s) 0.7535 − j 0.9887 (Rs + Rload + jX s ) X1m = 847 ⋅ (0.606 + j 0.795) ⋅ j ⋅ (0.018 + 5.734 + j14.4) = 911∠200.4° 14.4 The rotor electric power is Pr + jQr r = 3Vr Ir ∗ = 3.847 ⋅ 911∠159.6 = −2.381 MW + j 0.806 MVAR Again, the reactive power in the rotor is absorbed by the machine for magnetization, while reasonable power is delivered by the rotor The problem is that for f1 = 50 Hz and S = −0.05, the speed we are talking about is ω r = ω1(1 − s) = 1.05ω1 Should the speed be large, say corresponding to S = −0.25, the power delivered at maximum rotor voltage (874 V when stator is reduced and 847 × = 3388 V in reality), should be notably smaller than the value calculated; in fact, 16 times smaller The low reactive power required is due to the fact that the machine was designed with a high magnetization reactance (in P.U.; x1m = 5), and the slip is now reasonable ( S = −0.05 ) © 2006 by Taylor & Francis Group, LLC 1-28 Variable Speed Generators Stator Rotor w1 = const SG 3~ Diode rectifier on rotor w1 wr VsVariable (thyristor variac) Rf - field circuit resistance sLf - field circuit inductance to voltage Vs control FIGURE 1.17 Wound rotor induction generator (WRIG) as brushless exciter 1.8 Operation of WRIG in the Brushless Exciter Mode With a brushless exciter, the power is delivered through the rotor, after rectification, to the excitation circuit of a synchronous generator (Figure 1.17) The commutation in the diode rectifier causes harmonics in the rotor current, but for its fundamental, the power factor may be considered as unity The diode rectifier commutation causes some voltage reduction as already shown in Chapter in Synchronous Generators (the paragraph on excitation systems) The rotor rotates opposite to the stator mmf, and thus, ω = ω1 + ω r > ω1 (1.81) The frequency in the rotor is at its minimum at zero speed and then increases with speed If the WRIG is provided with a number of poles that is notably larger than that of the SG, then the frequency ω2 would be larger than ω1: ω = ω1 + 2πn1 p1 ; n1 = f1 pg (1.82) with pg-pole pairs in the SG with excitation that is fed from the WRIG exciter:  p  ω = ω1 1 +   pg    (1.83) The larger p1/pg, the higher the rotor (slip) frequency; with p1 /pg = 3, 4, good results may be obtained The WRIG-exciter is supplied through a static variac at constant frequency ω1, so the converter’s cost is low The machine equations (Equation 1.42) remain valid, but we will use ω2 instead of Sω1: I s Rs − V s = − jω1 Ψs = − jω1(Ls I s + L1m I r ) I r Rr + V r = − jω Ψr = − jω (Lr I r + L1m I s ) The speed ωr is now negative (ωr < 0), that is, ω1 > and ω2 > The slip S = ω /ω1 > © 2006 by Taylor & Francis Group, LLC (1.84) 1-29 Wound Rotor Induction Generators (WRIGs): Steady State We already used the positive sign (+) on the left side of rotor equation to have positive power for generating Also, for simplicity, a resistance load will be considered: V r = I r ⋅ Rloadr (1.85) The Equation 1.84 with Equation 1.85 may be solved simply for stator and rotor currents: Is = I r= j jI r (Rr + jω Lr + Rloads ) ω L1m (1.86) Vs (Rs + Rload + jω Lr )(Rs + jω1 Ls ) ω L1m (1.87) + jω1 Lm The electromagnetic torque Te is ( Te = p1 Re al j Ψs I s ∗ ) = 3p L 1m ( Re al jI s Ir ∗ ) (1.88) At zero speed, the WRIG-exciter works as a transformer, and all the active and reactive power is delivered by the stator When the speed increases — with resistive load in the rotor circuit — the stator “delivers” the reactive power to magnetize the machine and the active power to cover the losses and some part of the load active power The bulk of the active power to the load comes, however, from the mechanical power Pm The higher the ratio ω2/ω1, the higher the Pm contribution to Pr (rotor-delivered active power) Example 1.3: WRIG as Brushless Exciter Consider a WRIG with the main data: Rs = Rr = 0.015 P.U , Lsl = Lrl = 0.14 P.U , L1m = P.U.,VSNI = 440 V(star), I SN = 1000 A, the frequency f1 = 60 Hz , and the rotor speed nN = 1800 rpm The number of pole pairs is p1 = The rotor-to-stator turns ratio is ars = Determine the following: • The rotor frequency f2(ω2) and the ideal maximum no-load rotor voltage • The rotor-side load resistance voltage, current, power Pr0 at zero speed, and Ir = 1000 A in the rotor • The required stator voltage, current, and input active and reactive powers Ps, Qs, for the same load resistance Rload and current load Ir = 1000 A,but at nN = 1800 rpm Solution • The rotor-side frequency f2(ω2) is simply as follows (Equation 1.82): ω = ω1 + 2π n1 p1 ω1  2π ⋅ 1800 ⋅  60 = 1 +  = 4ω1 2π 60   So, f = f1 = 240 Hz The ideal no-load rotor voltage Vr r (unreduced to the stator, for full stator voltage at speed nN), is as follows: Vrr0 = ars ⋅ Vs ⋅ © 2006 by Taylor & Francis Group, LLC ω2 = ⋅ 440 ⋅ = 1760 V (line voltage , RMS) ω1 1-30 Variable Speed Generators The rotor circuit might be designed to comply with this voltage during an excitation 4/1 forcing At zero speed (ω2 = ω1), the ideal rotor voltage would be (V ) r r stall = ars ⋅ Vs ⋅ ω1 = ⋅ 440 ⋅ = 440 V ω1 • The machine parameters in Ω (all reduced to the stator) are as follows: Xn = VSNl / I SN = (440/ ) = 0.2543 Ω 1000 So, Rs = Rr = (Rs )P U ⋅ X n = 0.015 ⋅ 0.2543 = 3.8145 ⋅10−3 Ω Lsl = Lrl = ( X sl )P U ⋅ L1m = ( X1m )P U ⋅ Xn 0.2543 = 0.14 ⋅ = 9.45⋅10−5 H ⋅ 2π 60 ω1 Xn 0.2543 = 3⋅ = 2.0247 ⋅10−3 H 2π 60 ω1 At zero speed, ω2 = ω1, the rotor current Ir may be calculated from the following (Equation 1.87): (I r )ω = =ω1 = jV s ( Rs + Rload + jω Lr )( Rs + jω1Ls ) ω L1m + jω1 L1m (440/ ) (3.8145⋅10−3 + Rload + j 2π ⋅60⋅2.119⋅10−3 )(3.8145⋅10−3 + j 2π ⋅60⋅2.119⋅10−3 ) 2π 60⋅2.0247⋅10−3 + j 2π ⋅ 60 ⋅ 2.0247 ⋅10−3 Finally, Rload = 0.226 Ω So, the rotor voltage Vr (reduced to the stator) is Vr = Rload ⋅ Ir = 0.226 ⋅1000 = 226 V For voltage regulation, ∆V = Vs − Vr 254 − 226 = = 0.1102 = 11.02% Vs 254 The large leakage reactances of the stator and the rotor are responsible for this notable voltage drop (notable for a transformer or an induction machine, but small for an SG of any type) The rotor-delivered power Pr is as follows: Pr = 3Vr ⋅ Ir = 3x 220 ⋅1000 = 678 KW © 2006 by Taylor & Francis Group, LLC 1-31 Wound Rotor Induction Generators (WRIGs): Steady State • Now, we make use of Equation 1.87 to calculate the stator voltage required for Ir = 1000 A, with  (R + R + jω L ) ⋅ (R + jω L )  r 1m loadr s  V s= I r j  r ω L1m      (0.0038145 + 0.226 + j ⋅ 2π ⋅ 240 ⋅ 2.119 ⋅10−3 )(0.0038145 + j ⋅ 2π ⋅ 60 ⋅ 2.119 ⋅10−3 )   = 1000 j    2π ⋅ 240 ⋅ 2.0247 ⋅10−3   + 2π ⋅ 60 ⋅ 2.204 ⋅10−3 V s = 1000 j ⋅ (−0.0721 + j 0.06406) = −64.06 − j72.1 Vs = 96.8 V (RMS per phase) The stator I s from Equation 1.86 is I s = jI r (Rr + jω Lr + Rload ) ω L1m =j 1000 ⋅ (0.02298 + j 3.1938) = −1046.6 + j75.3; I s = 1049.3 A > Ir 3.0516 The stator active and reactive powers are as follows: ∗ Ps + jQs = 3V s I s = ⋅ (−64.06 − j72.1) ⋅ (−1046.6 − j75.3) Ps = 184.752 KW , Qs = 240.751 KVAR The delivered electric power through the rotor Pr is still 678 kW, as the load resistance and current were kept the same, but most of the power now comes from the shaft as Ps « Pr A few remarks are in order: • As the machine is rotated, less active power is delivered through the stator, with much of it “extracted” from the shaft (mechanically) This is a special advantage of this configuration • With the machine in motion (ωr = 3ω1), the required stator voltage decreases notably A static variac may be used to handle such a 1/5 voltage reduction easily • The machine magnetization is provided by the stator, and because ω2 = 4ω1, the power factor in the stator is poor • The magnetization by the stator is also illustrated by Is > I r • The stator voltage reserve at full speed may be used for forcing the excitation (load) current in the supplied synchronous machine excitation, but, in that case, the rotor voltage would increase above the rated value (440 V root mean squared [RMS]/line) The rotor winding insulation and the flying diode rectifier have to be sized for such events • The capability of the WRIG to serve as an exciter from zero speed — demonstrated in this example — makes it a good solution when the excitation power is required from zero speed, as is the case in variable-speed large synchronous motors or generators • The internal reactance of the WRIG is important to know in order to assess voltage regulation and to model the machine with rectified output • To emphasize the “synchronous” reactance of WRIG as an exciter, the stator current is eliminated from the stator equation by introducing the stator flux Ψ s : Ψr = © 2006 by Taylor & Francis Group, LLC Lm Ψ +L I Ls s sc r (1.89) 1-32 Variable Speed Generators The rotor equation (Equation 1.84) may be written now as follows: V r = − jω Lm Ψ − (Rr + jω Lsc )I r = E r − Z ex I r Ls s (1.90) Z ex = Rr + jω Lsc (1.91) The term Z ex represents the internal (synchronous) impedance of WRIG as an exciter source ∗ The first term in Equation 1.90 is the emf E r : E r = − jω Lm Ψ Ls s (1.92) The stator flux may be considered variable, with stator voltage as follows (Rs = 0): − jω1 Ψ s ≈ V s (1.93) Vs Lmω Lsω1 (1.94) Consequently, Er = An equivalent circuit based on Equation 1.90 and Equation 1.94 may be built (Figure 1.18) Basically, the emf E r ′ varies with ω2 (that is, with speed for constant ω1) and with the stator voltage Vs The “synchronous” reactance of the machine is, in fact, the short-circuit reactance So, the voltage regulation is reasonably small, and the transient response is expected to be swift; a definite asset for excitation control As the frequency in the rotor is large (ω2 > ω1), the core losses in the machine have to be considered One way to this is to “hang” a core resistance RFe in parallel with the emf E′ , RFe may be taken r as a constant, to be determined either from measured or calculated core losses PFe: pFe = Er' = Vs RFe Lmw Lsw1 Er′2 RFe jω2Lsc (1.95) Rr Ir Vr FIGURE 1.18 Equivalent circuit (phase) for wound rotor induction generator (WRIG) as an exciter source © 2006 by Taylor & Francis Group, LLC Wound Rotor Induction Generators (WRIGs): Steady State 1-33 1.9 Losses and Efficiency of WRIG The loss components in WRIG may be classified as follows: • • • • Stator-winding losses Stator core losses Rotor-winding losses Mechanical losses The stator-winding losses are due to alternative currents flowing into the stator windings With constant frequency (f1 = 50 (60) Hz), only in medium and large power machines is the skin effect important Roebel bars may be used in large power WRIGs to keep the influence of the skin effect coefficient below 0.33 (that is, 33% additional losses): ( s pcos = 3I s (Rs )dc ⋅ + K skin ) (1.96) In the rotor, the frequency f2 = Sf1, and with WRIGs, |f2| < 0.3f1 The rotor-to-stator turn ratio ars is chosen to be larger than 1(ars = 1/|Smax|) for low stator voltage WRIGs (up to to MW) and, in this case, the skin effect in the rotor is negligible However, in large machines, as the rotor voltage will probably not go over to kV (line voltage), even in the presence of specially built slip-rings, the rotor currents are large, in the range of thousands of amperes, again, transposed conductors are needed for the rotor windings There will be some skin effect, but, as the rotor frequency |f1| < 1/3f1, in general, its influence will be less important than in the stator (Krskin < K sskin < 0.3): r pcor = 3Ir21(Rr )dc ⋅ (1 + K skin ) (1.97) For details on skin effect, see Chapter in Synchronous Generators The fundamental stator core and rotor core losses may be approximated by an aggregated core-loss resistance RFe: RFe = RFes (ω1) + RFer (ω ) (1.98) This is exposed to the airgap emf Em: E m = − jX m Im ; Im = I s + Ir (1.99) So, pFes = 3( X m Im )2 RFe (ω1) (1.100) pFer = 3( X m Im )2 RFer (ω ) (1.101) The values of stator and rotor core loss resistances RFe and RFer may be obtained through experiments or from the design process When |ω2| < ω1, the rotor core losses are definitely smaller than in the stator This is not so when the WRIG is used as an exciter (|ω2| » ω1), and thus, even though the rotor core volume is larger in the stator, the rotor core losses are larger Additional losses occur in the stator and rotor windings in relation to the circuit time harmonics due (mainly) to the static power converter connected to the rotor They are strongly dependent on the PWM strategy and on the switching frequency © 2006 by Taylor & Francis Group, LLC 1-34 Variable Speed Generators Additional core losses occur due to space and time harmonics in the mmf of stator and rotor windings, in the presence of double slotting Current time harmonics bring additional core losses The additional space harmonics core losses occur on the rotor and stator surface toward the airgap N N Generally, only the first slot harmonics ν ss = p s ± 1, ν rs = p r ± 1, as influenced by the corresponding first1 order airgap magnetic conductance harmonics, are considered to produce surface core losses that deserve attention [2] Current time harmonics, on the other hand, produce additional core losses mainly along a thin layer along the slot walls Mechanical losses include ventilator (if any) losses, bearing-friction losses, brush-friction losses, and windage losses (Pmec): η= Ps + Pr Ps + Pr = Pm Ps + Pr + Σp (1.102) Σp = pcos + pcor + pFe + ps + pmec + psr For generating, Ps, in Equation 1.102, is always considered positive (delivered), while Pr is positive (delivered) for supersynchronous operation, and Pr < (absorbed) for subsynchronous operation Pm is the mechanical (input) power The slip-ring losses are denoted by psr, and the strayload losses are denoted by ps For details on efficiency (through iso-efficiency curves), see Reference [9] 1.10 Summary • WRIGs are provided with three-phase AC windings on the rotor and on the stator WRIGs are also referred to as DFIGs or DOIGs • WRIGs are capable of producing constant frequency ( f1) and voltage stator output power at variable speed if the rotor windings are controlled at variable frequency ( f2) and variable voltage The rotor frequency f2 is determined solely by speed n (rps) and f1:f2 = f1 − np1; p1 equals the number of pole pairs; and p1 is the same in the stator and in the rotor windings • WRIGs may operate as both motors and generators subsynchronously (n < f1/p1) and supersynchronously (n > f1/p1), provided the static power converter that supplies the rotor winding is capable of bidirectional power flow • The slip is defined as S = ω2/ω1 = f2/f1 and is positive for subsynchronous operation and negative for supersynchronous operation, and so is f2 Negative f2 means the opposite sequence of phases in the rotor is followed • WRIG is adequate in applications with limited speed control range (|Smax| < 0.2 to 0.3), as the rating of the rotor-side static converter is around PSN|Smax|, where PSN is the rated stator power The electric power Pr in the rotor is delivered for generating in supersynchronous operation and is absorbed in subsynchronous operation: Pr ≈ P∗SN S The total maximum power Pt delivered supersynchronously is thus, Pt = Ps + Pr = PSN (1+ | Smax |) • Consequently, in supersynchronous operation, the WRIG can produce significantly more total electric power than the rated power at synchronous speed (S = 0) • WRIG may also operate at synchronism, as a standard synchronous machine, provided the rotorside static power converter is able to handle DC power Back-to-back voltage source PWM converters are adequate for the scope It may be argued that, in this case (S = 0), a WRIG acts like a damperless SG True, but this apparent disadvantage is compensated for by the presence of fast close-loop control of active and reactive power, which produces the necessary damping any time the machine deviates from synchronism WRIG is also adequate to work as a synchronous condensator and contribute massively, when needed, to voltage control and stability in the power grid © 2006 by Taylor & Francis Group, LLC 1-35 Wound Rotor Induction Generators (WRIGs): Steady State • WRIG has laminated iron cores with uniform slots to host the AC windings Integer q (slots/pole/ phase) windings are used Open slots may be used only on one side of the airgap Axial cooling unistack cores are now in use up to to MW, while axial–radial cooling multistack cores are necessary above MW • To avoid parasitic synchronous torque, it suffices to have different numbers of slots in the rotor and the stator With large q and chorded coils, the main mmf harmonics are reduced and also reduced are their asynchronous parasitic torques • The rotor-to-stator turn ratio ars may be chosen as unity, but in this case, a voltage matching transformer is needed between the static converter in the rotor and the local power grid Alternatively, ars = 1/|Smax| > when the transformer is eliminated • A WRIG may be magnetized either from the stator or from the rotor, so the magnetization curve may be calculated (or measured) from both sides • When the reactive power is delivered through the rotor (overexcitation), the stator may operate at the unity power factor The lagging power factor in the stator seems to be a moderate to large burden on the rotor-side static converter kilovoltampere rating • A minimum kilovoltampere rating of the rotor-side static power converter is obtained when the stator power factor is leading (underexcitation Ψ r < Ψ s ) Ψr and Ψs are, respectively, the rotor and the stator flux linkage amplitudes per phase • For operation at the power grid, synchronization is required However, synchronization is much faster and easier than with SGs, because it may be performed at any speed ω r > ω1(1− | Smax |) by controlling the rotor-side converter in the synchronization mode to make the power grid and stator voltages of the WRIG equal to each other and in phase The whole synchronization process is short, as the rotor voltage and frequency (phase) are controlled quickly by the static power converter without any special intervention by the prime mover’s governor • The values of active power and reactive powers Ps, Qs, Pr, Qr, vs the rotor voltage (power) angle δ are somewhat similar to those in the case of cylindrical rotor SGs, but additional asynchronous power terms are present, and the stable operation zones depend heavily on the value and sign of slip (Figure 1.12) However, the decoupled active and reactive power control (see Chapter 2) eliminates such inconveniences to a great extent • The peak value of synchronous power components in Ps, Pr , for constant rotor flux, depend on the short-circuit reactance (impedance) of the machine Voltage regulation is moderate, for the same reason • The reactive power Qrr, absorbed from the rotor-side converter at f2 = S1f1 frequency, is “magnified” in the machine to the frequency f1, Qr = Qrr/|S|, as it has to produce the magnetic energy stored in the short-circuit and magnetization inductances Operation at unity power factor in the stator at full power leads, thus, to a moderate increase in rotor-side static converter kilovoltampere rating for |Smax| < 0.25 • The WRIG may also operate as a stand-alone generator It was demonstrated that such an operation is preferred for low reactive power requirements at low negative slips Constant frequency, constant voltage output in the stator with autonomous load does not seem to be advantageous when the speed varies by more than ±5% Ballast loads may be handled at any speed effectively, at smaller slip • With the stator short-circuited, the WRIG may be run as a motor to start the prime mover, say, for pumping in a pump-storage plant • After acceleration to ω r > ω1 ⋅ (1− | Smax |), the stator circuit is opened, the sequence of rotor voltages is changed, and their frequency f2 and amplitude are reduced to produce the conditions necessary for quick stator synchronization After that, motoring or generating operation is commanded subsynchronously or supersynchronously • The WRIG may operate in the brushless exciter mode to produce DC power on the rotor side with a diode rectifier and thus feed the excitation of a synchronous machine from zero speed up to the desired speed • The stator is supplied through a static voltage changer (soft starter type) at constant frequency ω ω1, while the rotor moves such that the rotor frequency ω = ω1 + | ω r |> ω1 With ω2 = 3, 4, good © 2006 by Taylor & Francis Group, LLC 1-36 Variable Speed Generators performance is obtained In all situations, the magnetization (reactive power) is delivered through the stator, but most of the load active power comes from the shaft mechanical power, and only a small part comes from the stator At zero speed, however, all the excitation power is delivered by the stator, electrically • When the speed increases, for constant rotor voltage, the stator voltage of the WRIG exciter is reduced considerably So, there is room for excitation forcing needs in the SG, provided the WRIG exciter insulation can handle the voltage The internal impedance of WRIG for brushless exciter mode is, again, the short-circuit impedance Thus, the commutation of diode reduction of the DC output voltage should be moderate • Besides fundamental winding and core losses, additional losses occur in the windings and magnetic cores of WRIGs due to space and time harmonics • The WRIG was proven to be reliable for delivering power at variable speed with very fast decoupled active and reactive control in industry up to 400 MW/unit It is yet to be seen if the WRIG will get a large share in the electric power generation of the future, at low, medium, and high powers per unit References T Kuwabara, A Shibuya, H Feruta, E Kita, and K Mitsuhashi, Design and dynamic response characteristics of 400 MW adjustable speed pump storage unit OHKAWACHI station, IEEE Trans., EC-11, 2, 1996, pp 376–384 I Boldea, and S.A Nasar, Induction Machine Handbook, CRC Press, Boca Raton, FL, 2001, p 327 I Boldea, and S.A Nasar, Electric Drives, CRC Press, Boca Raton, FL, 1998, chap 14 J Tscherdanze, Theory of double-fed induction machine, Archiv fur electrotechnik, 15, 1925, pp 257–263 (in German) A Leonhard, Asynchronous and synchronous running of the general doubly fed three phase machine, Archiv fur Electrotechnik, 30, 1936, pp 483–502 (in German) F.J Bradly, A mathematical model for the doubly-fed wound rotor generator, IEEE Trans., PAS103, 4, 1998, pp 798–802 M.S Vicatos, and J.A Tegopoulos, Steady state analysis of a doubly-fed induction generator under synchronous operation, IEEE Trans., EC-4, 3, 1989, pp 495–501 I Cadirci, and M Ermi, Double output induction generator operating at subsynchronous and supersynchronous speeds: steady state performance optimization and wind energy recovery, Proc IEE, 139B, 5, 1992, pp 429–442 A Masmoudi, A Toumi, M.B.A Kamoun, and M Poloujadoff, Power flow analysis and efficiency optimization of a doubly fed synchronous machine, EMPS J., 21, 4, 1993, pp 473–491 10 D.G Dorrel, Experimental behavior of unbalanced magnetic pull in three phase induction motors with excentric rotors and the relationship to teeth saturation, IEEE Trans., EC-14, 3, 1999, pp 304–309 © 2006 by Taylor & Francis Group, LLC ... stator output power at variable speed if the rotor windings are controlled at variable frequency ( f2) and variable voltage The rotor frequency f2 is determined solely by speed n (rps) and f1:f2... as an exciter from zero speed — demonstrated in this example — makes it a good solution when the excitation power is required from zero speed, as is the case in variable- speed large synchronous... constant frequency output, the rotor frequency ω2 has to be modified in step with the speed variation This way, variable speed at constant frequency (and voltage) may be maintained by controlling the